## Economics 440Public FinanceFall 2016Allen HeadAssignment 1(Due Friday, September 30th)1. Pure exchan

Economics 440Public FinanceFall 2016Allen HeadAssignment 1(Due Friday, September 30th)1. Pure exchange economies are completely described by specifications of agentsâ€™ preferencesand endowments. Consider the following two good, two agent pure exchange economy:â€¢ Let xhi denote agent hâ€™s consumption of good i, where i = 1, 2 and h = 1, 2.â€¢ Let ? h = (?1h , ?2h ) denote the quantities of goods x1 and x2 with which agent h isinitially endowed. Let ? = (? 1 , ? 2 ) be the aggregate endowment.Preferences:U 1 (x11 , x12 ) = 3×11 + 2×12(1)U 2 (x21 , x22 ) = x21 + 2×22(2)Endowments:?1 = (2, 4)?2 = (3, 4).(3)a. Draw an Edgeworth box diagram for this economy showing the set of Pareto efficientallocations.b. For one of the allocations in your diagram that is not Pareto efficient, identify inyour diagram (one of) the other feasible allocations Pareto superior to it. Explain.c. Is the endowment a Pareto efficient allocation? Why or why not?d. Do agents have equal marginal rates of substitution at the Pareto efficient allocations in this economy? If so, explain why this is important. If not, explain howthis can be.12. Consider the following economy with two agents and two consumption goods, one ofwhich is used to produce the other. Let Household hâ€™s consumption of the two goodsbe denoted xh1 and xh2 respectively, where h = 1, 2.Preferences:Household h has preferences given by:Uh (xh1 , xh2 ) = ln xh1 + 2 ln xh2h = 1, 2.Endowments:Both households are endowed with 2 units of good one (i.e. ?1h ). There is no endowmentof good two.Technology:Let Y2 denote the total amount of good two available to the economy. It is producedusing good one, by means of the following technology:14 ? (x11 + x21 )3where the term in brackets on the right-hand side of the equation above is the amountof good one that is used as an input to production of good two rather than consumed.Y2 =Consider the problem of a â€œsocial plannerâ€ who maximizes an equally weighted sumof agentsâ€™ utilities subject to the feasibility constraints.a. Write down the social welfare maximization problem that the planner solves.b. Use this problem to derive necessary conditions for an allocation to be Paretoefficient in this economy.c. Calculate the allocation that maximizes social welfare.d. Show how problems of this sort can be used to find all of the Pareto efficientallocations for this economy. Note: You do not have to find all of these allocations,just show how you could do so.